Question

NEED HELP ASAP!!!!

Find the exact value by using a half-angle identity.


sin five pi divided by twelve

Answer 1

`(1)/(2)\sqrt{2+√(3)}`

Explanation:

we know that

An half-angle identity is equal to

`sin((\theta)/(2))=(+/-)\sqrt{(1-cos(\theta))/(2)}`

we have

`sin((5\pi)/(12))`

The angle
`(5\pi)/(12)=75^o` ----> belong to the First Quadrant, so the value of the sine is positive

Let

`(\theta)/(2)=(5\pi)/(12)`

so

`{\theta=(5\pi)/(6)`

`sin((5\pi)/(12))=\sqrt{(1-cos(\theta))/(2)}`

`cos(\theta)=cos((5\pi)/(6))=cos(150^o)=-(√(3))/(2)`

substitute

`sin((5\pi)/(12))=\sqrt{(1-(-(√(3))/(2)))/(2)}`

`sin((5\pi)/(12))=\sqrt{(1+(√(3))/(2))/(2)}`

`sin((5\pi)/(12))=\sqrt{(2+√(3))/(4)`

`sin((5\pi)/(12))=(1)/(2)\sqrt{2+√(3)}`